Discrete and Computational Geometry, WS1415 Exercise Sheet “4”: Randomized Algorithms for
Geometric Structures IV
University of Bonn, Department of Computer Science I
• Written solutions have to be prepared until Tuesday 11th of Novem- ber 14:00 pm. There will be a letterbox in the LBH building.
• You may work in groups of at most two participants.
• Please contact Hilko Delonge, hilko.delonge@uni-bonn.de, if you want to participate and have not yet signed up for one of the exercise groups.
• If you are not yet subscribed to the mailing list, please do so at https://lists.iai.uni-bonn.de/mailman/listinfo.cgi/lc-dcgeom
Exercise 10: 3D Convex Hull by Conflict Lists 6 points Given a set N of n half-spaces each of which is defined by a hyperplane in the 3D space, a 3D convex hull H(N) of N is the common intersection of all half-spaces of N, Let S1, S2, . . . , Sn be a random sequence of N, and let Ni be {S1, S2, . . . , Si}. Please develop a randomized algorithm to construct H(N) by computingH(N4), H(N5), . . . , H(Nn) iteratively using the conflict lists. In other words, for i≥4, obtainH(Ni+1) from H(Ni) by addingSi+1. To simplify the description, we assume there is no four half-spaces whose defining hyperplanes intersect at the same point.
1. Define a configuration in H(Ni)
2. Define a conflict relation between a configuration in H(Ni) and a half- space in N =\Ni
3. Describe the insertion ofSi+1 using the conflict lists 4. Describe the updation of the conflict lists
5. Prove the complexity of this randomized incremental construction (Hint: Let cap(Si+1) be the intersection between edges of H(Ni) and the complement of Si+1. For an edge e of H(Ni+1) which does not belong to H(Ni), e and cap(Si+1) form a cycle, and if a half-space I ∈ S \ Ni+1 intersects e, I must intersect one of edges of the cycle except e.)
Exercise 11: 3D Convex Hull by History Graph 6 points Given a set N of n half-spaces each of which is defined by a hyperplane in the 3D space, a 3D convex hull H(N) of N is the common intersection of all half-spaces of N, Let S1, S2, . . . , Sn be a random sequence of N, and let Ni be {S1, S2, . . . , Si}. Please develop a randomized algorithm to construct H(N) by computingH(N4), H(N5), . . . , H(Nn) iteratively using the history graph. In other words, for i ≥ 4, obtain H(Ni+1) from H(Ni) by adding Si+1. To simplify the description, we assume there is no four half-spaces whose defining hyperplanes intersect at the same point.
1. Define a configuration in H(Ni)
2. Define the parent and child relation between a configuration inH(Ni)\
H(Ni+1) and a configuration in H(Ni+1)\H(Ni) 3. Describe the insertion ofSi+1 using the history graph
4. Prove the complexity of this randomized incremental construction (Hint:
• Let cap(Si+1) be the intersection between edges ofH(Ni) and the com- plement of Si+1. For an edge e of H(Ni+1) which does not belong to H(Ni), e and cap(Si+1) form a cycle, and if a half-space I ∈S\Ni+1 intersects e, I must intersect one of edges of the cycle except e.
• There are three kind of edges in H(Ni+1), and the last two belong to H(Ni+1)\H(Ni)
1. an edge is also an edge of H(Ni) 2. an edge is a part of an edge of H(Ni)
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3. an edge is completely new and contained in the hyperplane defin- ing Si+1
• There are two kind of edges in H(Ni)\H(Ni+1) 1. an edge is fully contained in Si+1.
2. an edge is only partially contained in Si+1, and the intersection between it and the complement ofSi+1 is an edge of H(Ni) (the second kind edge of H(Ni+1)).
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